On the Herbrand Notion of Consistency for Finitely Axiomatizable Fragments of Bounded Arithmetic Theories

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1 On the Herbrand Notion of Consistency for Finitely Axiomatizable Fragments of Bounded Arithmetic Theories Leszek Aleksander Ko lodziejczyk Institute of Mathematics, Warsaw University Banacha 2, Warszawa, Poland January 23, 2006 Abstract Modifying the methods of Z. Adamowicz s paper Herbrand consistency and bounded arithmetic (Fund. Math. 171 (2002)), we show that there exists a number n such that m S m (the union of the bounded arithmetic theories S m ) does not prove the Herbrand consistency of the finitely axiomatizable theory S n 3. From the point of view of bounded arithmetic, the concept of consistency based on Herbrand s theorem has at least two interesting features. Firstly, it has a more combinatorial flavour than standard consistency notions, and thus lends itself more naturally to combinatorial interpretations (cf. [Pud]). Secondly, it seems reasonably weak. It is well-known that for stronger concepts of consistency, such as ordinary Hilbert- or Gentzen-style consistency (Cons), or even bounded consistency (consistency with respect to proofs containing only bounded formulae, BdCons), there is typically a very large gap between a given theory and the theories whose consistency it can prove. Wilkie and Paris ([WP87]) Part of this work was carried out while the author was a Foundation for Polish Science (Fundacja na rzecz Nauki Polskiej) scholar. 1

2 proved the celebrated result that I 0 + Exp Cons(Q). For BdCons, the gap is somewhat smaller, but still significant: Pudlák ([Pud90]) proved that S 2 BdCons(S2), 1 and this was later improved by Buss and Ignjatović ([BI95]) to S 2 BdCons(S2 1 ), where S2 1 is axiomatized by BASIC plus two additional open axioms. Of course, the existence of such gaps leaves us no hope of using strong consistency notions to separate bounded arithmetic theories. For Herbrand consistency (HCons), on the other hand, the situation has seemed to be different. In fact, for quite a long time it was even open whether the analogue of Gödel s second incompleteness theorem for Herbrand consistency holds in the case of weak theories. A positive answer was finally given in a series of papers a few years ago. Gödel s theorem for HCons was proved for I 0 + Ω 2 by Adamowicz and Zbierski ([AZ01]), for I 0 + Ω 1 by Adamowicz ([Ada01]), and for a certain axiomatization of I 0 and a number of other theories by Willard ([Wil02]) 1. Another paper by Adamowicz, [Ada02], reproved the theorem for I 0 +Ω 2 (and generally I 0 +Ω m, m 2) using a different technique. In the present paper, we observe that a modification of the methods of [Ada02] leads to a considerably stronger result. It turns out that the gap phenomenon known for Cons and BdCons occurs to some degree also for HCons. More specifically, we prove: Main theorem. There exists n such that the theory m S m (or, equivalently, I 0 + m N Ω m) does not prove the Herbrand consistency of S n 3 (where S n 3 is defined like S n 2 but with the language expanded by a function symbol for # 3 ). The theorem can be proved with S n 2 instead of S n 3 if a symbol for the unary ω 1 function (equal to x#x) is added to the language. The reason why such a symbol makes a difference is that it allows some Herbrand proofs to become exponentially shorter. The importance of our result seems to be mainly negative: it rules out the possibility of separating some theories via a Gödel-style argument based on Herbrand consistency. It should be noted, however, that relativizations of Herbrand consistency to definable cuts might be able to prove separations 1 Willard s paper actually shows that I 0 does not prove its own consistency w.r.t. semantic tableau deduction, which is closely related, but not necessarily equivalent (in weak theories) to Herbrand consistency. This result does carry over to Herbrand deduction for some natural axiomatizations of I 0. However, to the author s knowledge it is still open whether it holds for all natural axiomatizations. 2

3 where the full notion is too strong. We touch upon this issue briefly towards the end of the paper. The paper is divided into six sections. The preliminary section 1 discusses some basic definitions and notation. In section 2, we formalize the notion of Herbrand consistency. In section 3, we explain how this formalized notion can be used to obtain new models for a theory. Section 4 contains the proof of our theorem, and section 5 outlines how to modify the proof in order to obtain the result for S n 2. We conclude with some additional remarks in section 6. The framework developed in sections 2 and 3, along with some of the reasoning from section 4 (in particular, the proof of Lemma 4.2), is essentially a reconstruction of parts of [Ada02]. Nevertheless, we try to provide a reasonable amount of detail. This makes the paper more self-contained, and additionally, it helps us discuss some of the subtleties involved see sections 5 and 6. 1 Preliminaries We assume a general familiarity with bounded arithmetic, in particular, with the meaning of the symbols I 0, I 0 + Exp, Σ b n, Π b n, BASIC, S2 n, S 2, and with notions such as that of a (possibly definable) cut or initial segment of a model. [HP93] or [Kra95] may serve as an introduction. L PA is the basic language of Peano Arithmetic (we take it to consist of 0, 1, +,, and ). We write L BA to denote the standard language of the bounded arithmetic theory S 2. The symbols 0, Σ 1, and Π 1 have their usual meanings as names for certain classes of L PA -formulae. In particular, 0 is the class of bounded L PA -formulae. BΣ 1 is the well-known collection schema for Σ 1 formulae. For m 3, the language L m is formed by expanding L BA with binary function symbols # 3,..., # m. The intended meaning of these symbols is defined inductively: x# m+1 y is 2 x #m y for m 2, where # 2 is just the usual smash function #. For each m 3, the theory Sm n is an L m -analogue of S2 n, with the axioms x# k y = x # k 1 y + 1 (k = 3,..., m), 3

4 added to fix the meaning of the new symbols. Like S2 n, all of the Sm n theories are finitely axiomatizable for n 1. The L m -analogue of S 2 is called S m. Variants of the S m theories may also be formulated in L PA. The standard way to do this is to use unary functions called ω m, where ω m (x) = x# m+1 x for m 1. The graph of each ω m is 0 -definable in I 0, and Ω m is the sentence x y y = ω m (x). It is easy to see that the theory I 0 + Ω m corresponds very closely to S m+1 (more precisely, the latter is a conservative extension of the former). Some notational conventions: if f is a unary function, then a superscript, as in f k, always denotes iteration. We sometimes write exp to denote the exponential function 2 x (especially when dealing with iterations). The symbol 2 k is an abbreviation for exp k 1 (2). The symbol log represents both the obvious function and the cut of logarithmically small elements, i.e. those x for which 2 x exists. Note that if our language contains L BA, the cut log can be defined by the particularly simple formula y x = y. For k 2, the initial segment log k (which is a cut whenever # k 1 is a total function) consists of the elements for which exp k exists. If a model M is given, the symbols log M and log k M stand for the corresponding cuts or segments interpreted in M. If I is any initial segment, ri is the initial segment determined by elements of I multiplied by r. Similarly, ω m (I) is the segment determined by elements of the form ω m (i) for i I. If a is an element of some model, a N is the cut determined by the standard powers of a. Likewise, if f is an increasing function, f N (a) is the cut determined by the standard iterations of f applied to a. A bar, as in x, always indicates a tuple. If I is a cut, then x I ϕ is shorthand for a formula stating that there exists x all of whose elements are in I such that ϕ holds. Finally, a word on sequence coding. Throughout the paper, we may work with any of the standard feasible coding schemes commonly used in bounded arithmetic when L BA is available. However, at one point in the proof of Lemma 4.2 it will be most convenient to use the coding given by Gödel s β-function. Recall that a β-code for a sequence s 1,..., s k is a pair of appropriately chosen numbers a, b, where b has, among others, the property that all of the numbers b(j +1)+1 for j = 0,..., k are pairwise coprime. Each s j is recovered from a, b as the remainder of the division of a by b(j +1)+1. 4

5 In other words, c is the j-th element of the sequence whose β-code is a, b if d (a = d(b(j + 1) + 1) + c & c < b(j + 1) + 1). (1) One helpful feature of β-codes is that that the formula in (1) is very simple once d is given, it is actually an open formula. We also note that any bounded and logarithmically long sequence (in the sense of some standard coding) has a β-code. 2 Herbrand consistency via evaluations Let ϕ( x) be a formula of the form x 1 y 1... x k y k ϕ(x 1, y 1,..., x k, y k ), where ϕ is open (any formula can be transformed into a logically equivalent formula of this shape by taking the prenex normal form and adding superfluous quantifiers). The Herbrandization of ϕ, He(ϕ), is defined as x 1... x k ϕ(x 1, f 1 (x 1 ),..., x k, f k (x 1,..., x k )), where the f i s are new function symbols, called Herbrand functions. Notice that Herbrandization is dual to Skolemization, i.e. He(ϕ) is the negation of the Skolemization of ϕ (modulo the de Morgan laws). Obviously, ϕ is provable (in first-order logic) if and only if He(ϕ) is provable. Herbrand s theorem states that ϕ is provable if and only if a finite disjunction of formulae of the form (+) ϕ(t 1, f 1 (t 1 ),..., t k, f k (t 1,..., t k )), where the t i s are terms of the language extended by adding Herbrand functions, is a propositional tautology. Any formula as in (+) is called an Herbrand variant of ϕ, while a tautological disjunction of Herbrand variants is sometimes called an Herbrand proof of ϕ. Now let T be a finitely axiomatizable arithmetical theory, say, T = {τ 1,..., τ r }. We shall call T Herbrand inconsistent if there is an Herbrand proof of τ 1... τ r. T is Herbrand consistent otherwise. By Herbrand s theorem, the Herbrand consistency of a theory is equivalent to consistency in the usual Hilbert-style sense. However, in weak theories 5

6 which do not prove Herbrand s theorem, these two concepts may well differ. In this case, the class of weak theories includes even I 0 + Exp. Indeed, there are theories, such as Robinson s Q and I 0, whose Herbrand consistency is provable in I 0 + Exp but whose Hilbert-style consistency is not. In the remainder of this section, we develop the formalization of Herbrand consistency used in the present paper. Assume T is as above. We expand the language of T, L T, by adding Herbrand functions f1, j..., f j i j of the appropriate arity for each axiom τ j of T (of course, the number i j depends on the number of quantifiers in τ j ). Call this extended language L H T. Call the set of closed LH T -terms Term(LH T ). L H T is a finite language, in the sense that it contains only a finite number of constant individual, predicate, and function symbols. Hence, we may gödelize L H T in such a way that a closed LH T -term of length l will have a Gödel number of about c l, where c is a fixed standard natural number. Thus, the Gödel number of an L H T -term of the form f(t 1,..., t k ) will be polynomial in the numbers for t 1,..., t k, and the length lh( ) of a term will be linearly related to the length of its Gödel number in the sense of the function. From now on, we identify elements of Term(L H T ) with their Gödel numbers. Consider any interval of the form [0, a], where a log. The simple atomic sentences over [0, a] are the sentences obtained by substituting elements of Term(L H T ) [0, a] for variables in formulae of the form x 1 = x 2, R(x 1,..., x k ), or f(x 1,..., x k ) = x k+1, where R, f are symbols of L T. An evaluation over [0, a] is any assignment p of truth values (0 or 1) to (all) simple atomic sentences over [0, a] which respects the equality axioms. We will write p = ψ for p (ψ) = 1. The code of an evaluation over [0, a] is roughly exponential in the number of simple atomic sentences over [0, a], which in turn is polynomial in a. So, the formalization of evaluations works well in any theory which proves the totality of # (and thus the closure of log under multiplication). An evaluation over [0, a] can be extended in the obvious way to arbitrary open sentences whose terms are from [0, a]. This allows us to define the concept of a T -evaluation: p is a T -evaluation over [0, a] if p = ψ for any ψ obtained by substituting terms from [0, a] into the open part of the Skolemization of some τ j (where we treat the f j i s as Skolem functions)2. A weak form of the Herbrand consistency of T may now be expressed by 2 Naturally, we only consider those ψ in which all the terms, not just the ones substituted for variables, belong to [0, a]. 6

7 the following sentence HCons (T ): for any a log, there exists a T -evaluation over [0, a]. To see that HCons is implied by any adequate formulation of Herbrand consistency, consider its negation. HCons (T ) means that for some a, there is no T -evaluation over [0, a]. In other words, there is no way to assign truth values to atomic formulae over [0, a] in such a way that all instances of the Skolemizations of axioms of T become true. But since an instance of the Skolemization of τ j is precisely the negation of an Herbrand variant of τ j, this means that the disjunction of all Herbrand variants of the τ j s involving terms from [0, a] is a propositional tautology. Thus, T is Herbrand inconsistent. Since HCons only mentions Herbrand disjunctions of relatively small terms (the terms are to be from log), the above argument does not exclude the possibility that in some cases HCons is essentially weaker than full Herbrand consistency. In this paper, however, we are aiming for results on the unprovability of Herbrand consistency, so this is not a problem. Remark. Concerning the requirement that an evaluation respect the equality axioms: this corresponds to treating equality axioms as part of the theory (or of logic), and is normally needed to ensure that models obtained from T -evaluations (see the next section) are well-defined and satisfy T. Actually, however, for the purposes of the present paper this requirement is merely convenient, not strictly necessary. For theories in the language L BA (containing BASIC), the reason is that one of the BASIC axioms characterizes x = y as x y &y x, and other axioms imply that all the operations of L BA respect equality in the sense of this characterization. The # m functions for m > 2 might not satisfy the equality axioms exactly, as the axioms only define them up to. Nevertheless, a model obtained from an Sm-evaluation, n m > 2, is a well-defined model for S2 n and can be expanded in a canonical way to satisfy the totality of # m. This expansion also satisfies Sm. n 3 Models from evaluations In this section, we discuss how the existence of T -evaluations over appropriately large intervals can be used to construct models of T. The basic idea behind the construction seems to date back to Lemma 3.3 of [Pud85]. 7

8 Let a log M, where M is a (nonstandard) model of some amount of arithmetic, and let p be a T -evaluation on [0, b], where b log M, b > a N. We may use p to define a model M(p, a) as follows. Define the relation = p on the set Term(L H T ) [0, b] by: t 1 = t 2 iff p = t 1 = t 2. Since p respects the equality axioms, this is an equivalence relation. The universe of M(p, a) is (Term(L H T ) an )/ =p. The relations and operations are defined in the natural way: t 1 t 2 holds in M(p, a) iff p = t 1 t 2 ; [t 1 ] + [t 2 ] is [t 1 + t 2 ], etc. This is correct as, firstly, = p is a congruence with respect to all the operations (equality axioms again), and secondly, the closure of a N under multiplication guarantees that all the operations can be defined (recall from section 2 that for any function symbol f of L T and even L H T, the term f(t 1,..., t k ) is just polynomially larger than the terms t 1,..., t k )). Obviously, for any open L T -formula ϕ, M(p, a) = ϕ([a 1 ],..., [a k ]) iff p = ϕ(a 1,..., a k ). In particular, since p is a T -evaluation, M(p, a) satisfies each instance of the Skolemization of any τ j in which terms from Term(L H T ) an are substituted for the universally quantified variables. But since all elements of M(p, a) are given by terms from Term(L H T ) an, this simply means that M(p, a) satisfies any τ j, and hence, M(p, a) = T. We will now show that if T contains BASIC, then the model M(p, a) has another important property: the cut a N in M is isomorphic to an initial segment of M(p, a), at least with respect to the basic arithmetical language of addition and multiplication. To define the isomorphism, we need to have canonical terms naming the elements of a N. It is natural to choose the well-known dyadic numerals, defined inductively as follows. The numerals 0, 1, 2 are the terms 0, 1, and 1+1, respectively (we do not write parentheses explicitly unless it is necessary); 2k is k 2 for k 2; and 2k + 1 is 2k + 1 for k 1. It should be noted that k is a term of logarithmic length w.r.t. k, so it is bounded by k c for some fixed standard c. Thus, the numerals for elements of a N are themselves in a N. Lemma 3.1. If M = S 1 2 and T contains BASIC, then a N is isomorphic (w.r.t. the language L PA ) to an initial segment of M(p, a). 8

9 Proof. Of course, it is enough to show that the interval [0, a] is isomorphic to an initial segment of M(p, a) (with addition and multiplication treated as relations since they are not total). The isomorphism will map each i a to [i]. To prove that this really is an isomorphism, we will need to use induction (in M) a number of times. In each case, it will be clear that S 1 2 suffices for the inductive arguments. We first prove that for each i < a, p = i + 1 = i + 1. The proof is by induction, with the inductive step split into two cases. The case where i = 2k for some k is obvious, so we only deal with the case where i = 2k + 1. In this case i + 1 is, by definition, k By BASIC, so by the inductive assumption, p = i + 1 = k k + 1, p = i + 1 = (k + 1) + (k + 1). On the other hand, i + 1 is (2k + 1) + 1, so by BASIC, p = i + 1 = ((k + k) + 1) + 1. Since the axioms of BASIC include the associativity and commutativity of addition, it follows that p = i + 1 = i + 1. The next step is to show that for i, j a, p = i + j = i + j (this actually holds even if i + j > a). But once we know that p identifies j + 1 with j + 1, this is a very straightforward inductive argument which uses the two axioms x + 0 = x and x + (y + 1) = (x + y) + 1 contained in BASIC. Similar arguments show that an analogous fact holds for multiplication, and that the mapping given by i [i] is injective and preserves the ordering. Altogether, we have obtained an isomorphic embedding of [0, a] into M(p, a). It remains to prove that its range is an initial segment. To this end, we will show the following: for any i a and any term t, if p = t i, then there is a j i such that p = t = j. The argument is by induction on i. The base step uses the BASIC axioms 0 x and (x y & y x) x = y. In the inductive step, assuming p = t i + 1, we use BASIC to get p = t = i + 1 t + 1 i

10 If the first disjunct holds, then we are done, if it is the second, then the axiom x + y x + z y z and some work in BASIC gives p = t i, so we may use the inductive assumption. The following immediate corollary will play an important role in the next section (see the proof of Lemma 4.2). Corollary 3.2. If M = S 1 2 and T contains BASIC, then for any 0 formula θ and any a 1,..., a k a, M(p, a) = θ([a 1 ],..., [a k ]) iff M = θ(a 1,..., a k ). 4 The results In the present section, we prove our main theorem. We will first prove a somewhat weaker version, Theorem 4.1, and then discuss how to modify the proof in order to obtain the stronger result. Theorem 4.1. For every m 3 there exists an n such that S m does not prove the Herbrand consistency of S n m. We start by proving an adaptation of a crucial technical lemma from [Ada02] (the proof is also adapted from [Ada02]). The original version states, in essence, that if a bounded arithmetic theory T proved its own Herbrand consistency, then any appropriately small witness for a 0 formula in a model of T could be made still logarithmically smaller in another model of T. The observation we need here is that a similar witness reduction phenomenon occurs also when we consider the provability of Herbrand consistency for one theory in a possibly different theory. Lemma 4.2. Let m 3, n 1, k 0, and assume that Sm n+k HCons (Sm). n Then for any 0 formula θ( x) such that Sm n+k + x log m θ( x) is consistent, Sm n + x log m+1 θ( x) is also consistent. Proof. Assume S n+k m S n+k m HCons (Sm) n and consider a 0 formula θ( x) such that + x log m θ( x) is consistent. Take any nonstandard model M = Sm n+k which contains a tuple ā = a 1,..., a k such that M = θ(ā) and all elements of ā are in log m. We will use the fact that M = HCons (S n m) to construct a model of S n m in which x log m+1 θ( x) holds. 10

11 Assume w.l.o.g. that a 1 is maximal among a 1,..., a k. Take any b log M greater than (2 2a 1 ) N (such a number b must exist, since 2 2a 1 is in log M, which is closed at least under #). By HCons (S n m), there exists an S n m-evaluation p on [0, b]. Consider the model M(p, 2 2a 1 ). By the previous section, M(p, 2 2a 1 ) = S n m. Moreover, by Corollary 3.2, M(p, 2 2a 1 ) = θ([a 1 ],..., [a k ]). So, all we need to show is that the elements [a 1 ],..., [a k ] are contained in log m+1 M(p, 2 2a 1 ). Naturally, we may restrict ourselves to [a 1 ]. In all models of S 1 m and for all elements a, exp m+1 (a) is roughly equal to 2 m # m... # m 2 m }{{} 2 a whenever these two values exist (actually, the latter is somewhat larger, as # m is defined in terms of the function, whose values are a bit larger than the real binary logarithm). So, it is enough to prove that in M(p, 2 2a 1 ), there exists 2 m # m... # m 2 }{{ m. } 2 a 1 The term t := 2 a 1 {}}{ 2 m # m... # m 2 m certainly is an element of (2 2a 1 ) N (to see this, just note that the length of t is linear in 2 a 1 2 2a 1 ). Thus, we just have to check that the relation between [a 1 ] and [t] in M(p, 2 2a 1 ) is indeed as it should be. Working in M(p, 2 2a 1 ), consider the sequence where i is the largest number 2 a 1 2 m, 2 m # m 2 m,..., 2 m # m... # m 2 }{{ m, } [i] such that 2 m # m... # m 2 }{{ m [t] } [i] (note that by Lemma 3.1, we can use induction in M(p, 2 2a 1 ) to find the largest element 2 a 1 satisfying this condition). This sequence contains only 11

12 elements below [t] and is at most logarithmically long, so it has a β-code in M(p, 2 2a 1 ). Let this β-code be [t A ], [t B ], where t A and t B are terms from Term(L H S ) 1 m n (22a ) N (in M). For any j i, there are some t j, t j in Term(L H S ) 1 m n (22a ) N such that [t j] witnesses that [t j ] is the [j]-th element of the sequence whose β-code is [t A ], [t B ], i.e. M(p, 2 2a 1 ) satisfies [t A ] = [t j] ([t B ] ([j] + 1) + 1) + [t j ] & [t j ] < [t B ] ([j] + 1) + 1. Since the above is an open formula, in M we have p = t A = t j (t B (j + 1) + 1) + t j & t j < t B (j + 1) + 1. (2) For each j i, consider the smallest pair t j, t j Term(L H S ) 1 m n (22a ) N for which (2) holds (already S2 1 is enough to do this, as we just need to find the smallest appropriate pair t j, t j below b, and b log M). An easy induction in M shows that for all j i, p = t j = In particular, p identifies t i with the term j {}}{ 2 m # m... # m 2 m. i {}}{ 2 m # m... # m 2 m. We are now ready to show that i = 2 a 1, which will complete the proof. Suppose the contrary, i.e. i < 2 a 1. Then i a 1, and hence it is easy to check that i+1 {}}{ p = 2 m # m... # m 2 m t. i {}}{ 2 m # m... # m 2 m # m 2 m are identical, there- The terms fore i+1 {}}{ 2 m # m... # m 2 m and p = t i # m 2 m t, so M(p, 2 2a 1 ) satisfies [t i ]# m 2 m [t]. But [t i ] is equal to 2 m # m... # m 2 m }{{} [i] for the largest i 2 a 1 for which this is not greater than [t]. Contradiction. 12

13 We may now employ Lemma 4.2 to prove Theorem 4.1 in the following way. Proof of Theorem 4.1. Assume that: ( ) for some m 3, S m proves HCons (S n m) for all n. By compactness, it follows from ( ) that there exists an increasing sequence n k : k N of natural numbers such that n 0 1 and for each k, S n k+1 m HCons (S n k m ). Having fixed such a sequence, we will use Lemma 4.2 to derive the following: ( ) for every 0 formula θ( x), if S m + x log m θ( x) is consistent, then I 0 + Exp + x θ( x) is consistent. The statement ( ) is known to be false (see the proof of theorem 1.1 in [Ada02] or theorem V.5.36 in [HP93]). Therefore, showing that ( ) implies ( ) will complete the proof of the theorem. Thus, let us consider a 0 formula θ( x) such that S m + x log m θ( x) is consistent. We claim that: ( ) for any l 1, S 1 m + x log m+l θ( x) is consistent. Indeed, fix l and consider S n l m. Since S n l m + x log m θ( x) is consistent and S n l m HCons (S n l 1 m ), Lemma 4.2 implies that S n l 1 m + x log m+1 θ( x) is consistent. It follows that S n l 1 m + y log m x log y θ( x) is also consistent. Since S n l 1 m HCons (S n l 2 m ) and x log y θ( x) is 0, we may use Lemma 4.2 again to obtain the consistency of S n l 2 m + y log m+1 x log y θ( x), or, equivalently, of S n l 2 m + x log m+2 θ( x). Iterating this procedure l times, we will eventually show that S n 0 m + x log m+l θ( x) is consistent. This completes the proof of claim ( ), as n 0 was assumed to be at least 1. Applying compactness to ( ), we see that the theory T := S 1 m + θ( d) + { d log l : l N} is consistent (where d is a tuple of new constants of the appropriate length). Consider any M = T. M is a model for Sm, 1 and it contains a tuple d satisfying θ such that all standard iterations of the exponential function on elements of d exist. Let M be the submodel of M determined by the cut exp N (max( d)). Clearly, M = Sm 1 + Exp + θ( d). But the theory Sm 1 + Exp is equal to I 0 + Exp, except for the difference in language (this is essentially established by the standard proof of the finite axiomatizability of I 0 +Exp), 13

14 so M is a model for I 0 + Exp + x θ( x). This concludes the proof of ( ), and hence also of Theorem 4.1. The proof of the main theorem is quite similar to the one given above, but it requires two additional observations. The first of these is that an analogue of Lemma 4.2 still holds if the two theories involved differ not only in the amount of induction they prove, but also in the growth-rate of functions they make total 3. This can be shown in the same way as Lemma 4.2. Lemma 4.3. Let m 3, n 1, k, l 0, and assume S n+k m+l HCons (Sm). n Then for any 0 formula θ( x) such that S n+k m+l + x log3 θ( x) is consistent, Sm n + x log 4 θ( x) is also consistent. Proof idea. The proof is entirely analogous to that of Lemma 4.2 for the case of m = 3. In particular, the term 2 a 1 {}}{ 16# 3... # 3 16 is used (where a 1 has the same meaning as in the proof of Lemma 4.2). Now the second observation: if we are able to make a witness for some formula exponentially smaller in a model of S n 3, then we can also make it almost exponentially smaller in a model of S n m for some higher m: Lemma 4.4. The family of theories {S n m} n,m 1 has the following properties: (i) Let m 3, n 1, k 0, and assume Sm n+k HCons (S3 n ). Then for any 0 formula θ( x) such that Sm n+k + x log 3 θ( x) is consistent, Sm n + x ω m 4 (log 4 ) θ( x) is also consistent 4. (ii) Let m 3, n 1, k, l 0, and assume that Sm n+k+l HCons (S3 n+k ) and Sm n+k HCons (S3 n ). Then for any 0 formula θ( x) such that Sm n+k+l + x log 3 θ( x) is consistent, Sm n + x log 4 θ( x) is also consistent. 3 I owe the idea that a version of Lemma 4.2 should be true in this situation to Konrad Zdanowski. 4 In both the lemma and the proof, we are using the convention that ω 0 (x) = x 2, ω 1 (x) = 2x, and ω 2 (x) = x

15 Proof. Assume the hypothesis of part (i) and consider θ( x) 0 such that Sm n+k + x log 3 θ( x) is consistent. By Lemma 4.3, there exists a (nonstandard) model M = S3 n containing a tuple d such that max( d) log 4 M and M = θ( d). Let d := max( d). W.l.o.g. we may consider only the case where d is nonstandard. In that case, there is some c M such that ω m 4 (c) d, but ωm 5(c) N < d. Indeed: already S2 1 proves that there exists a smallest number c such that ω m 4 (c) d. It can also be proved that there is a value of ω m 4 between d and ω m 5 (ω m 5 ((d)). Thus, ω m 4 (c) ω m 5 (ω m 5 ((d)), and since ω m 4 dominates all standard iterations of ω m 5 for all nonstandard arguments, it follows that ωm 5(c) N < d. We have ωm 1(exp N 4 (c)) < exp 4 (d), as ω m 1 (exp 4 (x)) is approximately exp 4 (ω m 5 (x)). Let M be the submodel of M determined by ωm 1(exp N 4 (c)). Clearly, M = S3 n, and since ω m 1, and thus # m, is total in M, we also get M = Sm. n Furthermore, M = θ( d). Finally, d ω m 4 (log 4 M ), because d ω m 4 (c), and c is obviously in log 4 M. Hence, part (i) of the lemma is proved. Now let m, n, k, l be as required for part (ii) and let θ( x) be such that Sm n+k+l + x log 3 θ( x) is consistent. We will apply part (i) twice. The first application, with n := n + k and k := l, proves that Sm n+k is consistent with x ω m 4 (log 4 ) θ( x). In the second application, taking θ(y) to be x ω m 4 (log y) θ( x), we get the consistency of y ω m 4 (log 4 ) x ω m 4 (log y) θ( x) with S n m. This means that in some model of S n m, a tuple of witnesses for θ( x) may be found in ω m 4 (log(ω m 4 (log 4 ))), or equivalently in ω m 4 (ω m 5 (log 5 )). But in any model of S n m, ω m 4 (ω m 5 (log 5 )) is contained in log 4, so in fact we may find the tuple of witnesses in log 4. Proof of Main theorem. Assume that m S m does prove HCons (S3 n ) for all n. In that case, there exists an increasing sequence n k : k N such that n 0 3 and for each k, there exists a number n [n k, n k+1 ] such that S n k+1 n k+1 HCons (S3 n ) and Sn n k+1 HCons (S n k 3 ). 5 Fix such a sequence. We now emulate the proof of Theorem 4.1, using Lemma 4.4(ii) instead of Lemma 4.2, to get 5 The sequence n k : k N can be defined inductively. Let n 0 be any number greater or equal to 3. Given n k, let ñ k n k be any number such that Sñk ñ k HCons (S n k 3 ), and let n k+1 ñ k be such that S n k+1 n k+1 HCons (Sñk 3 ). Obviously, ñ k may play the role of n. 15

16 ( ) for any 0 formula θ( x), if m S m + x log 3 θ( x) is consistent, then I 0 + Exp + x θ( x) is consistent. All that remains is to show that ( ) is false. To see this, one may apply an argument analogous to the one used to disprove the statement ( ) (from the proof of Theorem 4.1) in the proof of theorem 1.1 in [Ada02]. One small modification is needed: we have to show that any model M of I 0 + Exp + BΣ 1 has a proper end-extension to a model of I 0 + m N Ω m. But this is easy: take the proper end-extension of M to a model M = I 0 guaranteed to exist by [WP89], and let M be the submodel of M determined by the cut {a M : ω m (m) (a) exists for all standard m}. M is clearly an end-extension of M to a model of I 0 + m N Ω m, and it is straightforward to show that the extension is proper 6. 5 The case of S n 2 One would like to extend our main theorem, or at least Theorem 4.1, to the most important case of m = 2. Unfortunately, there is a technical problem here. An extension of Lemma 4.2 to m = 2 would involve pushing witnesses down from log 2 to log 3. But the term required to do this, 2 a {}}{ 4#... #4, is polynomially related to 2 2a. Thus, we would need the model M(p, 2 2a ), which may be unavailable as 2 2a will in general fall outside log. Interestingly, however, even our main result does extend to m = 2 if the vocabulary is modified by adding a function symbol for ω 1. The benefit of having ω 1 is that it is a unary function, so it can be used to define fast-growing functions using shorter terms. In particular, for any x, ω1(x) a x#... #x, }{{} 2 a so exp 3 (a) is roughly equal to ω a 1(4). The size of the term ω a 1(4) is about 2 a, which is in log for a log 2. With this in mind, one may reconstruct the argument of section 4 to prove: 6 This particularly simple way of getting end-extensions to models of I 0 + m N Ω m was suggested to me by Zofia Adamowicz. 16

17 Theorem 5.1. Let S n, 2 be defined by adding the axiom ω 1 (x) = x#x to S n 2. Then there exists n such that m S m does not prove HCons (S n, 2 ). In what follows, we will briefly explain the details of the reconstruction. The reader uninterested in those details may prefer to skip the rest of the present section, save perhaps the remark at the very end. We want to proceed as in the proof of our main theorem. Therefore, we need appropriately modified versions of Lemmata 4.3 and 4.4. The analogue of Lemma 4.3 is as follows: Lemma 5.2. Let m 2, n 1, k 0, and assume Sm n+k HCons (S n, 2 ). Then for any 0 formula θ( x) such that Sm n+k + x log 2 θ( x) is consistent, S2 n + x log 3 θ( x) is also consistent. Proof idea. The reasoning is again based on the idea used in the proof of Lemma 4.2. This time, we need to move a tuple ā of witnesses for θ( x) down from log 2 in a nonstandard model M of Sm n+k into log 3 in a model of S2 n. To achieve this, we work with the term ω a 1 1 (4) (where the significance of a 1 is as before). As noted above, this term is roughly of the size 2 a 1, and (2 a 1 ) N is contained in log M, which allows the proof to go through as previously if m 3. In the case of m = 2 (which will not actually be needed for the proof of Theorem 5.1), there is an additional technicality. Here, the cut log is in general only closed under multiplication. Thus, it might happen that (2 a 1 ) N is cofinal in log M, leaving no room to find some log M b > (2 a 1 ) N and an evaluation p over [0, b] which could be used to define M(p, 2 a 1 ). Fortunately, there is an easy way out: a standard compactness argument shows that we may consider only models in which log is closed under taking some nonstandard power. If M is such a model, then obviously (2 a 1 ) N can no longer be cofinal in log M. The modified version of Lemma 4.4 is: Lemma 5.3. The family of theories {S n m} n,m 1 has the following properties: (i) Let m 2, n 1, k 0, and assume Sm n+k HCons (S n, 2 ). Then for any 0 formula θ( x) such that Sm n+k + x log 2 θ( x) is consistent, S2 n + x ω m 3 (log 3 ) θ( x) is also consistent. 17

18 (ii) Let m 2, n 1, k, l 0, and assume that Sm n+k+l HCons (S n+k, 2 ) and Sm n+k HCons (S n, 2 ). Then for any 0 formula θ( x) such that Sm n+k+l + x log 2 θ( x) is consistent, Sm n + x log 3 θ( x) is also consistent. Proof idea. Little changes in comparison with the proof of Lemma 4.4, except that we now use Lemma 5.2 instead of Lemma 4.3. For this reason, each log r appearing in the proof of Lemma 4.4 now becomes log r 1, ω m 4 becomes ω m 3, and ω m 5 becomes ω m 4 (so there is no need to define ω 3 ). Given Lemma 5.3(ii), we conclude the proof of Theorem 5.1 just like the proof of the main theorem in section 4. Of course, the statement ( ) has to be suitably modified. Remark. Note that instead of expanding the language by adding ω 1, we could have extended our theories by adding the seemingly irrelevant axiom x y y = x#x. Indeed, this axiom gives rise to a unary Herbrand function 7 which behaves as ω 1, so for some n, m S m does not prove HCons (S n 2 + x y y = x#x). Thus, the notion of Herbrand consistency, or at least our way of proving its unprovability, seems to be somewhat less robust than one could wish. 6 Concluding remarks We conclude the paper with two further remarks. One of these relates our results to an important open problem in the area of weak arithmetics, while the other concerns the notion of cut-free consistency. I. There is a famous open problem whether I 0 + Ω k is Π 1 -conservative over I 0 + Ω m for k > m. Our results indicate that the generally expected negative answer will not be shown by proving I 0 + Ω k HCons(I 0 + Ω m ), 7 If one defines the Herbrand consistency of a finitely axiomatizable theory T as the non-existence of an Herbrand proof for the prenex normal form of T (note that our approach, described in section 2, is slightly different), then the question whether this Herbrand function is unary or not will depend on the details of how exactly the normal form is built; more precisely, it will depend on the order of quantifiers in the prefix. 18

19 at least not if m 1. However, there can still be hope that the study of variants of Herbrand consistency might have some relevance for the Π 1 -conservativity problem, if a more careful quantitative analysis is involved. For any cut I, define the statement HCons dp(i) (T ) to be: For any set of terms A log closed under subterms and containing only terms whose depth is in I, there exists a T -evaluation over A. (Note that we are now allowing evaluations over sets which are not intervals of the form [0, a]; the depth of an L H T -term is defined inductively in the natural way.) HCons dp(i) is Π 1 whenever I has a Σ 1 definition, and it is well-known ([Pud85]) that any reasonable finitely axiomatizable T proves HCons dp(i) (T ) for some definable cut I. One possible way to solve the Π 1 -conservativity problem is to gain more insight into the question of exactly how large I can be for various bounded arithmetic theories. To give a specific example, let us focus on the case of I 0 + Ω 3 versus I 0 + Ω 2, or equivalently, S 4 versus S 3. We know from Theorem 4.1 that for some n, S 3 does not prove HCons (S3 n ). On the other hand, it can be shown that for any n and any fixed standard r, S 4 does prove HCons dp(r log4) (S n 3 ) ([Ada] 8 ). So, if Theorem 4.1 for m = 3 could be improved to the result that for some n and r, S 3 HCons dp(r log4) (S n 3 ), it would follow that I 0 + Ω 3 is not Π 1 conservative over I 0 + Ω 2. In this context, it is quite interesting that the methods of [AZ01] can be refined to show that for each sufficiently large n, S n 3 HCons dp(3 log4) (S n 3 ). So, the question to ask is whether the unprovability of HCons(S n 3 ), for large n, is as deep in S 3 as it is in S n 3. A positive answer would give a partial solution to the Π 1 -conservativity problem, while a negative answer would separate S 3 from its finitely axiomatizable fragments. 8 The basic idea of the argument is that in a model M in which ω 3 is total, ω 2 can be iterated at least log 4 times. Therefore, an evaluation on terms of depth from log 4 can be defined by interpreting them as elements of M. This idea can be elaborated to show that S 4 proves HCons dp(log4) (S n 3 ) for every n. Finally, Ω 3 implies that log 4 is closed under addition. 19

20 II. There are many notions of consistency similar in strength to Herbrand consistency. Most widely known among these is probably the concept of cutfree consistency (CFCons), i.e. consistency relative to cut-free proofs in the sequent calculus. Hence, it is natural to ask whether our theorems can be translated into results about the unprovability of CFCons. Herbrand consistency and cut-free consistency are known to be equivalent in I 0 +Exp. Moreover, analysis of the proof of this equivalence (as presented in [HP93]) reveals that already in S 2, any formula ϕ which has an Herbrand proof contained in log also has a cut-free proof. We have shown that for some n, models of m S m may contain Herbrand proofs of the inconsistency of S3 n. Unfortunately, these Herbrand proofs are comparable in size to evaluations over an interval [0, b], where b is larger than 2 2a for a log 3. Such evaluations are in general too large to fit inside log. Thus, our reasoning does not suffice to conclude the unprovability of CFCons(S3 n ) in m S m. The situation changes once we move to fragments of S 4. A proof that m S m does not prove HCons(S4 n ) for some n can be based on moving witnesses for 0 formulae not from log 3 to log 4, but from log 4 to log 5. The evaluations needed to do this only have to cover an interval of the form [0, b], where b > (2 2a ) N for some a log 4 (cf. the proof of Lemma 4.2 for m = 4). By a compactness argument, b itself can be pushed into log 2, which means that evaluations over [0, b] will be in log. Using this, one can show that a model of m S m may contain an Herbrand proof of the inconsistency of S4 n inside log. Consequently, a model of m S m may contain a cut-free proof of the inconsistency of S4 n. We have thus sketched a justification of the following theorem: Theorem 6.1. There exists n such that m S m CFCons(S n 4 ). Although the general algorithm for constructing a cut-free proof from an Herbrand proof results in an exponential blow-up in size, Herbrand proofs typically tend to be larger than cut-free proofs. For this reason, we conjecture that Theorem 6.1 may be improved to (at least) the unprovability of CFCons(S n 2 ) for some n in m S m. Acknowledgements. During the work on this paper, I have benefited greatly from comments by, and discussions with, Zofia Adamowicz and Konrad Zdanowski. I am very grateful to them both. I would also like to thank Dan Willard for explanations concerning the exact strength of his results from 20

21 [Wil02], and the anonymous referee for comments which allowed me to improve the structure of the paper and notice a small error in one of the proofs in the original draft version. References [Ada] Z. Adamowicz, private communication. [Ada01] [Ada02], On tableaux consistency in weak theories, Preprint 618, Institute of Mathematics of the Polish Academy of Sciences, 2001., Herbrand consistency and bounded arithmetic, Fundamenta Mathematicae 171 (2002), [AZ01] [BI95] [HP93] Z. Adamowicz and P. Zbierski, On Herbrand consistency in weak arithmetic, Archive for Mathematical Logic 40 (2001), S. R. Buss and A. Ignjatović, Unprovability of consistency statements in fragments of bounded arithmetic, Annals of Pure and Applied Logic 74 (1995), P. Hájek and P. Pudlák, Metamathematics of first-order arithmetic, Springer-Verlag, [Kra95] J. Krajíček, Bounded arithmetic, propositional logic, and complexity theory, Cambridge University Press, [Pud] [Pud85] [Pud90] P. Pudlák, Consistency and games, to appear in Proceedings of Logic Colloquium 2003., Cuts, consistency statements, and interpretations, Journal of Symbolic Logic 50 (1985), , A note on bounded arithmetic, Fundamenta Mathematicae 136 (1990), [Wil02] D. E. Willard, How to extend the semantic tableaux and cut-free versions of the second incompleteness theorem almost to Robinson s arithmetic Q, Journal of Symbolic Logic 67 (2002),

22 [WP87] A. J. Wilkie and J. B. Paris, On the scheme of induction for bounded arithmetic formulas, Annals of Pure and Applied Logic 35 (1987), [WP89], On the existence of end-extensions of models of bounded induction, Logic, Methodology, and Philosophy of Science VIII (Moscow 1987) (J.E. Fenstad, I.T. Frolov, and R. Hilpinen, eds.), North- Holland, 1989, pp